Almost optimal sliding mode control for linear time varying systems

This paper presents a new sliding mode control (SMC) design approach using embedding process based on measure theory. In this method, a state trajectory starting from a given initial point reaches a definite point on a sliding surface in the minimum finite time, and then tends to the origin along the sliding surface (SS). A measure theory approach with embedding process is used to solve such a problem in two phases. In the first phase, using an appropriate measure, an embedding is constructed to solve a time optimal control problem such that the system trajectory reaches the SS in minimum time, and in the second phase, using measure theory, a control is designed such that the system trajectory tends to the origin along the SS. The proposed SMC is piecewise continuous and chattering free.